On a cycle partition problem

Let GG be any graph and let c(G)c(G) denote the circumference of GG. We conjecture that for every pair c1,c2c1,c2 of positive integers satisfying c1+c2=c(G)c1+c2=c(G), the vertex set of GG admits a partition into two sets V1V1 and V2V2, such that ViVi induces a graph of circumference at most cici, i=1,2i=1,2. We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free graphs, certain important classes of perfect graphs, and graphs without too many intersecting long cycles, satisfy the conjecture.This work is inspired by a well-known, long-standing, analogous conjecture involving paths.